Extracting Local Manifold Geometry from Pretrained Diffusion Models in One Inverse Step

We propose \emph{DiffusionGeometryProbe}, a self-contained framework that, in a single inverse step of a pretrained denoising diffusion probabilistic model (DDPM), extracts a calibrated profile of the local geometry of the data manifold the model has internalised. Our contributions are: (i) a fixed-point reformulation of DDIM inversion whose contraction rate $\rho_g(t)$ admits closed-form bounds in $\sigma_{\max}(J_{\eps})$, enabling Banach- and Smale-$\alpha$-theory-based certification of any inversion scheme (Picard, Anderson, Newton-Raphson); (ii) a unifying probe class that returns, in a single pass, four mutually-corroborative local intrinsic-dimension (LID) estimators (FLIPD, Stanczuk, Yeats, local-PCA), the score-Jacobian condition number, a Cheeger-style spectral-gap proxy, a Witten-deformation Morse-basin count, and a {\L}ojasiewicz exponent of the score energy; (iii) a complete derivation of each measurement from a classical or recent result in differential / spectral / Morse geometry, with all spectral computations using only Jacobian--vector products and Hutchinson trace estimators, so the probe scales linearly in $D$. We validate the framework on a synthetic point-cloud DDPM trained on five $3$-manifolds of known topology, on the pretrained CIFAR-10 DDPM ($D{=}3072$) and on the pretrained CelebA-HQ-256 DDPM ($D{=}196{,}608$).